Composite \(\sqrt{2}\) subdivision surfaces. (English) Zbl 1171.65342

Summary: This paper presents a new unified framework for subdivisions based on a \(\sqrt{2}\) splitting operator, the so-called composite \(\sqrt{2}\) subdivision. The composite subdivision scheme generalizes 4-direction box spline surfaces for processing irregular quadrilateral meshes and is realized through various atomic operators. Several well-known subdivisions based on \(\sqrt{2}\) splitting operator and based on 1-4 splitting operator for quadrilateral meshes are properly included in the newly proposed unified scheme. Typical examples include the midedge and 4-8 subdivisions based on the \(\sqrt{2}\) splitting operator that are now special cases of the unified scheme as the simplest dual and primal subdivisions, respectively. Variants of Catmull-Clark and Doo-Sabin subdivisions based on the 1-4 splitting operator also fall in the proposed unified framework. Furthermore, unified subdivisions as extension of tensor-product B-spline surfaces also become a subset of the proposed unified subdivision scheme. In addition, Kobbelt interpolatory subdivision can also be included into the unified framework using VV-type (vertex to vertex type) averaging operators.


65D17 Computer-aided design (modeling of curves and surfaces)
Full Text: DOI


[1] Bertram, M., Biorthogonal loop-subdivision wavelets, Computing, 72, 29-39, (2004) · Zbl 1060.65017
[2] Catmull, E.; Clark, J., Recursively generated B-spline surfaces on arbitrary topological meshes, Computer-aided design, 10, 6, 350-355, (1978)
[3] Cohen, E.; Lyche, T.; Riesenfeld, R., Discrete b-splines and subdivision techniques in computer aided geometric design and computer graphics, Computer graphics and image processing, 14, 2, 87-111, (1980)
[4] de Boor, C.; Hollig, K.; Riemenschneiger, S., Box splines, (1993), Springer New York
[5] Doo, D.; Sabin, M., Behaviour of recursive division surfaces near extraordinary points, Computer-aided design, 10, 6, 356-360, (1978)
[6] Habib, A.; Warren, J., Edge and vertex insertion for a class of subdivision surfaces, Computer aided geometric design, 16, 4, 223-247, (1999) · Zbl 0916.68151
[7] Kobbelt, L., Interpolatory subdivision on open quadrilateral nets with arbitrary topology, Proceedings of EUROGRAPHICS 1996, Computer graphics forum, 15, 3, 409-410, (1996)
[8] Kobbelt, L., 2000. \(\sqrt{3}\)-Subdivision. In: SIGGRAPH 2000, pp. 103-112
[9] Lane, J.M.; Riesenfeld, R.F., A theoretical development for the computer generation and display of piecewise polynomial surfaces, IEEE transactions on pattern analysis and machine intelligence, 2, 1, 35-46, (1980) · Zbl 0436.68063
[10] Li, G.; Ma, W.; Bao, H., \(\sqrt{2}\) subdivision for quadrilateral meshes, The visual computer, 20, 2-3, 180-198, (2004)
[11] Li, G., Ma, W., Bao, H., 2004b. Interpolatory \(\sqrt{2}\)-subdivision surfaces. In: Proceedings of Geometric Modeling and Processing 2004, pp. 180-189
[12] Li, G., Ma, W., 2006. Adaptive unified subdivisions with sharp features. Preprint
[13] Li, G.; Ma, W., A method for constructing interpolatory subdivision schemes and blending subdivisions, Computer graphics forum, 26, 2, 185-201, (2007)
[14] Maillot, J.; Stam, J., A unified subdivision scheme for polygonal modeling, Eurographics 2001, Computer graphics forum, 20, 3, 471-479, (2001)
[15] Oswald, P.; Schröder, P., Composite primal/dual \(\sqrt{3}\) subdivision schemes, Computer aided geometric design, 20, 2, 135-164, (2003) · Zbl 1069.65562
[16] Oswald, P., Designing composite triangular subdivision schemes, Computer aided geometric design, 22, 7, 659-679, (2005) · Zbl 1119.65314
[17] Peters, J.; Reif, U., The simplest subdivision scheme for smoothing polyhedra, ACM transactions on graphics, 16, 4, 420-431, (1997)
[18] Prautzsch, H.; Boehm, W.; Paluszny, M., Bézier and B-spline techniques, (2002), Springer Berlin · Zbl 1033.65008
[19] Sovakar, A.; Kobbelt, L., API design for adaptive subdivision schemes, Computer & graphics, 28, 1, 67-72, (2004)
[20] Stam, J., On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree, Computer aided geometric design, 18, 5, 383-396, (2001) · Zbl 0970.68184
[21] Stam, J.; Loop, C., Quad/triangle subdivision, Computer graphics forum, 22, 1, 79-85, (2003)
[22] Sweldens, W., The lifting scheme: a custom-design construction of biorthogonal wavelets, Applied and computational harmonic analysis, 3, 186-200, (1996) · Zbl 0874.65104
[23] Velho, L., Using semi-regular 4-8 meshes for subdivision surfaces, Journal of graphics tool, 5, 3, 35-47, (2000)
[24] Velho, L., 2003. Stellar subdivision grammars. In: Eurographics Symposium on Geometry Processing, pp. 188-199
[25] Velho, L.; Zorin, D., 4-8 subdivision, Computer aided geometric design, 18, 5, 397-427, (2001) · Zbl 0969.68157
[26] Wang, H.; Qin, K.; Tang, K., Efficient wavelet construction with catmull – clark subdivision, The visual computer, 22, 9-11, 874-884, (2006)
[27] Warren, J.; Schaefer, S., A factored approach to subdivision surfaces, IEEE computer graphics & applications, 24, 3, 74-81, (2004)
[28] Warren, J.; Weimer, H., Subdivision methods for geometric design: A constructive approach, (2002), Morgan Kaufmann Publisher San Francisco
[29] Zorin, D., Schröder, P., 2000. Subdivision for modeling and animation. In: SIGGRAPH 2000, Course Note
[30] Zorin, D.; Schröder, P., A unified framework for primal/dual quadrilateral subdivision scheme, Computer aided geometric design, 18, 5, 429-454, (2001) · Zbl 0969.68155
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.